Optimal. Leaf size=78 \[ -\frac {\sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {\text {Ci}\left (\cos ^{-1}(a x)\right )}{6 a}+\frac {x}{6 \cos ^{-1}(a x)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4622, 4720, 4724, 3302} \[ -\frac {\sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {\text {CosIntegral}\left (\cos ^{-1}(a x)\right )}{6 a}+\frac {x}{6 \cos ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4622
Rule 4720
Rule 4724
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a x)^4} \, dx &=\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {1}{3} a \int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {x}{6 \cos ^{-1}(a x)^2}-\frac {1}{6} \int \frac {1}{\cos ^{-1}(a x)^2} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {x}{6 \cos ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}-\frac {1}{6} a \int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)} \, dx\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {x}{6 \cos ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{6 a}\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac {x}{6 \cos ^{-1}(a x)^2}-\frac {\sqrt {1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac {\text {Ci}\left (\cos ^{-1}(a x)\right )}{6 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 71, normalized size = 0.91 \[ \frac {2 \sqrt {1-a^2 x^2}-\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2+\cos ^{-1}(a x)^3 \text {Ci}\left (\cos ^{-1}(a x)\right )+a x \cos ^{-1}(a x)}{6 a \cos ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arccos \left (a x\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 66, normalized size = 0.85 \[ \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{6 \, a} + \frac {x}{6 \, \arccos \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{6 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a \arccos \left (a x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 0.81 \[ \frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arccos \left (a x \right )^{3}}+\frac {a x}{6 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arccos \left (a x \right )}+\frac {\Ci \left (\arccos \left (a x \right )\right )}{6}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3} \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x}{{\left (a^{2} x^{2} - 1\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}\,{d x} + a x \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right ) - \sqrt {a x + 1} \sqrt {-a x + 1} {\left (\arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} - 2\right )}}{6 \, a \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {acos}^{4}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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